1966 AHSME Problems/Problem 38
Problem
In triangle the medians
and
to sides
and
, respectively, intersect in point
.
is the midpoint of side
, and
intersects
in
. If the area of triangle
is
, then the area of triangle
is:
Solution
Construct triangle with points
being the midpoints of sides
, respectively. Proceed by drawing all medians. Then draw all medians (so draw
). Next, draw line
and label
's intersection with
as the point
. From the problem, the area of
is
, but by vertical angles we know that
. Furthermore, since line
is drawn from the midpoint of
to the midpoint of
, we know that
is parallel to
(via SAS similarity on triangles PCM and ABC). From these parallel lines we know that
which indicates that
. The linear ratio from
to
is 1:2 because line segment
is one half of line segment
since
and
make up the median
. Thus the area ratio is 1:4. So
has area
. Since
has the same height and base as
we know that the area of
. The medians form 3 triangles each with area
of the total triangle (these triangles are
). Thus since
.
- LitJamal
See also
1966 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 37 |
Followed by Problem 39 | |
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